The main claim of [1]} is that question:bhh has a positive answer if \((E,\nabla )\)  has irreducible monodromy and the genus of \(X\)  is at least 2. However, the following results answer question:bhh in the negative, even in this case. See remark:bhh-error for a discussion of the errors in previous claims that question:bhh had a positive answer.
